Intrinsic diophantine approximation in Carnot groups and in the Siegel model of the Heisenberg group

Abstract

We study an intrinsic notion of Diophantine approximation on a rational Carnot group G. If G has Hausdorff dimension Q, we show that its Diophantine exponent is equal to \((Q+1)/Q\), generalizing the case \(G=\mathbb {R}^n\). We furthermore obtain a precise asymptotic on the count of rational approximations. We then focus on the case of the Heisenberg group \(\mathbf {H}^n\), distinguishing between two notions of Diophantine approximation by rational points in \(\mathbf {H}^n\): Carnot Diophantine approximation and Siegel Diophantine approximation. We provide a direct proof that the Siegel Diophantine exponent of \(\mathbf {H}^1\) is equal to 1, confirming the general result of Hersonsky-Paulin, and then provide a link between Siegel Diophantine approximation, Heisenberg continued fractions, and geodesics in the Picard modular surface. We conclude by showing that Carnot and Siegel approximation are qualitatively different: Siegel-badly approximable points are Schmidt winning in any complete Ahlfors regular subset of \(\mathbf {H}^n\), while the set of Carnot-badly approximable points does not have this property.

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Acknowledgements

Joseph Vandehey was supported in part by the NSF RTG grant DMS-1344994. Anton Lukyanenko was supported by NSF RTG grant DMS-1045119. Part of the research was conducted by both authors while at MSRI, with support of the GEAR Network (NSF RNMS grants DMS 1107452, 1107263, 1107367). The authors would like to thank Ralf Spatzier for discussions about this paper.

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Correspondence to Anton Lukyanenko.

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Communicated by Shrikrishna G Dani.

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Lukyanenko, A., Vandehey, J. Intrinsic diophantine approximation in Carnot groups and in the Siegel model of the Heisenberg group. Monatsh Math 192, 651–676 (2020). https://doi.org/10.1007/s00605-020-01406-7

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Keywords

  • Badly approximable
  • Carnot group
  • Continued fractions
  • Diophantine approximation
  • Heisenberg group
  • Schmidt games

Mathematics Subject Classification

  • Primary 11J83
  • Secondary 22E25
  • 11J70
  • 53C17